Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lebesgue integral of a_p

  1. Mar 24, 2006 #1
    Let define the function:

    [tex] a_{p}(x)= 1 [/tex] if x is an integer and prime and 0 elsewhere, my

    question is.........what would be its Lebesgue integral let,s say from [c,d] with c and d positive and real..
     
  2. jcsd
  3. Mar 24, 2006 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Since this function is only non-zero on only a finite number of points between c and d, isn't the integral obviously 0?
     
  4. Mar 24, 2006 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It's Riemann (and hence lebesgue) integral is rather trivially zero on any interval. My question is why would you need to ask this?
     
  5. Mar 25, 2006 #4
    then why the integral of the function [tex] f(x)=1 [/tex] iff x is rational and 0 elsewhere is different from 0?....
     
  6. Mar 25, 2006 #5

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    You're sayig the Lebesgue integral of this function is non zero? How do you figure?
     
  7. Mar 25, 2006 #6

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    1 on a set of measure zero 0 every where else, aka almost everywhere zero. that the integral is zero of such a thing is practically the point of lebesgue theory.
     
  8. Mar 25, 2006 #7

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    It isn't! Who told you that it was? The lebesque integral of the function you give is 0 over any finite interval.

    IF, instead, you define f(x)= 1 if x is irrational and 0 if x is rational (1- your f(x)) then the integral of f over the interval [a, b] is b-a.
     
  9. Apr 2, 2006 #8
    i know i have posted this topic or analogue before but i have the doubts with lebesgue integration:

    a) the Lebesgue integral of exp(x)..is equal to Riemann integral of exp(x)

    b) [tex] D_{t}\int_{0}^{t}d\mu{f}= f ? [/tex]

    c)what would be the formula for integration by parts in Lebesgue integration?..

    thanks.
     
  10. Apr 2, 2006 #9

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    integration by parts requires the integrands to be differentiable or to be a derivative and hence continuous, so there is no point in using lebesgue integration, is there?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Lebesgue integral of a_p
  1. Integral closure (Replies: 20)

  2. Integral of a matrix (Replies: 4)

  3. Integrate matrix (Replies: 4)

  4. Integral Domain (Replies: 3)

Loading...